Looks like a psychological question with philosophical implications!
Difficult to say for certain, but I would venture that the difference comes down to the nature of the reward. Unless a child is a professional video game player, playing the game has no rewards beyond the satisfaction of playing. The child is moved to play by intrinsic motivation . In contrast, in most cases, the rewards of completing one's homework are largely extrinsic , stemming from the praise a child gets from parents or teachers for completing it, grades, etc. And there's an abundance of evidence that the nature of our motivation for engaging in an activity shapes how rewarding or worthwhile we find it: Activities we perform for their own sake are experienced as more rewarding or worthwhile than activities whose rewards lie outside it (in social esteem, income, and so on). This is why most of us find play more gratifying than work, even when play is hard work! Edward Deci https://en.wikipedia.org/wiki/Edward_L._Deci has...

Unfortunately, this is a tricky question for Kantian ethics to address.
On its face, it might appear that procreation (bringing a child into existence) in order to advance one’s own happiness treats the child merely as a means: One ‘uses’ the child to promote one’s own happiness.
But things get more complicated once we attend to exactly what this Kantian imperative says. The Kantian moral requirement you mention states that we are not to treat “humanity” merely as a means. There are debates as to exactly what Kant had in mind by “humanity” but the standard view is that “humanity” means the capacity for rational agency — the ability to choose our ‘ends’ (our goals or objectives) and the best means to those ends. But a newborn lacks “humanity” in this sense; it cannot choose ends for itself, etc. Nor can a fetus. All the more, a child who does not yet exist does not have humanity! Hence, it would appear that the apparent answer to your question is ‘no’: You cannot treat someone’s capacity for...

Second question first: Of course not! If 'talking back' means picking arguments with a teacher, that's not very productive -- or very philosophically minded. That said, I think many philosophers would agree that too much of formal education emphasizes the memorization or assimilation of 'established' knowledge as the expense of the sort of curiosity and questioning found in philosophy. There's a worldwide movement to promote philosophy education for children. Here are some good resources on that front:
http://depts.washington.edu/nwcenter/
http://p4c.com/
As to your first question: I don't have any empirical data to support this -- to my knowledge, how philosophers raise their children has never been studied. All the same , I would not at all be surprised to learn that many of the traits that one needs to be successful in philosophy -- a sense of puzzlement, attention to reasoning, comfort with uncertainty, respect for those with whom one disagrees -- are passed on by philosophers to their children. I...

Your questions touches on a number of issues within the emerging philosophical field of procreative ethics , the field addressing questions concerning the ethics of reproduction and parenting.
I concur with the spirit of your last sentence: It is interesting that landmark legal rulings in the United States establishing legal rights to use birth control and the right to abortion both appealed to the right to privacy. But if there is a right to procreate, it is probably not best modelled on a right to privacy.
Your comments about licensing parents echo a well-known argument given by Hugh LaFollette in a 1980 article (http://www.hughlafollette.com/papers/licensing.parents.pdf). Here's my reconstruction of LaFollette's argument:
1. Incompetent parenting is harmful to children.
2. Societies are justified in restricting access to activities that are potentially harmful to others if those restrictions significantly reduce the likelihood of harm from those activities. (Compare, for example, driving...